Value at Risk (VaR)

Sukrit Mittal Franklin Templeton Investments

Outline

Motivation: why downside risk?
Quantiles and loss distributions
Definition of Value at Risk
Measuring downside risk
Computing VaR: parametric method
VaR with normal returns – examples
VaR in the Black–Scholes model
Mathematical properties and proofs
Limitations of VaR
Exercises

1. Motivation: Why Downside Risk?

Variance treats upside and downside symmetrically.

Investors do not.

Losses hurt more than gains help.

Risk management therefore focuses on:

How bad can things get?

VaR is the industry’s first systematic answer.

From Portfolio Choice to Risk Management

So far:

We optimized portfolios
We priced assets

Now:

We measure losses
We control exposure

This shift is practical, not philosophical.

2. Quantiles and Loss Distributions

Let $L$ denote portfolio loss over a fixed horizon.

$L$ is a random variable.

Risk is encoded in its distribution.

Quantiles summarize extreme outcomes.

Quantiles

For $\alpha \in (0,1)$, the $\alpha$-quantile $q_\alpha$ satisfies:

\[\mathbb{P}(L \le q_\alpha) = \alpha\]

Interpretation:

With probability $\alpha$, losses do not exceed $q_\alpha$.

3. Definition of Value at Risk

The Value at Risk at level $\alpha$ is:

\[\text{VaR}_\alpha = \inf { x : \mathbb{P}(L \le x) \ge \alpha }\]

In words:

The worst loss not exceeded with probability $\alpha$.

Interpretation

Typical levels: 95%, 99%
Time horizon: 1 day, 10 days, 1 year

VaR answers a narrow question.

It does not describe tail severity beyond the quantile.

4. Measuring Downside Risk

VaR focuses on:

Left tail of returns
Right tail of losses

It ignores upside outcomes entirely.

This is a feature, not a bug.

Loss vs Return Convention

Let $R$ be portfolio return.

Define loss as:

\[L = -R\]

Quantiles of $L$ correspond to lower-tail quantiles of $R$.

Sign conventions matter.

5. Parametric VaR

Assume returns are normally distributed:

\[R \sim \mathcal{N}(\mu, \sigma^2)\]

Then losses $L = -R$ are also normal.

VaR becomes analytical.

Normal VaR Formula

Let $z_\alpha$ be the $\alpha$-quantile of the standard normal.

Then:

\[\text{VaR}*\alpha = -(\mu + \sigma z*{1-\alpha})\]

Often written as:

\[\text{VaR}*\alpha = \sigma z*\alpha - \mu\]

depending on conventions.

6. Numerical Example

Suppose:

$\mu = 0.1%$ daily
$\sigma = 1%$ daily
$\alpha = 99%$

Then $z_{0.99} \approx 2.33$.

\[\text{VaR}_{0.99} \approx 2.33% - 0.1% = 2.23%\]

This is a one-day VaR.

7. VaR in the Black–Scholes Model

Assume asset price follows:

\[\frac{dS_t}{S_t} = \mu dt + \sigma dW_t\]

Log-returns are normally distributed.

This aligns perfectly with parametric VaR.

Distribution of Log-Returns

Over horizon $T$:

\[\ln \frac{S_T}{S_0} \sim \mathcal{N}\left((\mu - \tfrac{1}{2}\sigma^2)T, \sigma^2 T\right)\]

Losses can be expressed explicitly.

Closed-form VaR follows.

VaR for a Stock Position

For investment value $V_0$:

\[\text{VaR}*\alpha = V_0 \left(1 - e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma \sqrt{T} z*{1-\alpha}} \right)\]

This is exact under Black–Scholes assumptions.

8. Mathematical Properties and Proofs

Proposition

VaR is positively homogeneous:

\[\text{VaR}*\alpha(cL) = c,\text{VaR}*\alpha(L), \quad c>0\]

Proof follows directly from quantile scaling.

Lack of Subadditivity

In general:

\[\text{VaR}*\alpha(L_1 + L_2) \nleq \text{VaR}*\alpha(L_1) + \text{VaR}_\alpha(L_2)\]

VaR may penalize diversification.

This is not a technicality.

It is a fundamental flaw.

9. Limitations of VaR

Ignores tail severity
Not coherent in general
Sensitive to distributional assumptions

Yet:

VaR remains a regulatory standard.

History matters.

10. Exercises

Exercise 1

Compute 95% and 99% VaR for a portfolio with:

$\mu = 5%$ annually
$\sigma = 20%$ annually

Assume normality.

Exercise 2

Derive the Black–Scholes VaR formula step by step.

Identify each probabilistic assumption.

Exercise 3

Construct an example where diversification increases VaR.

Explain the intuition.

Final Takeaways

VaR is a quantile-based risk measure
It focuses exclusively on downside risk
It is analytically convenient
It is theoretically imperfect

Next: coherent risk measures.

Because regulators learned the hard way.